From e24c67450d38ffdde84b1191a85f9c4b1687d054 Mon Sep 17 00:00:00 2001
From: FloppaInspector <136809316+FloppaInspector@users.noreply.github.com>
Date: Sun, 6 Apr 2025 00:39:54 +0900
Subject: [PATCH] Update knuth-optimization.md

In time complexity proof cancellation explanation, j=N -> j=N-1
---
 src/dynamic_programming/knuth-optimization.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/dynamic_programming/knuth-optimization.md b/src/dynamic_programming/knuth-optimization.md
index 535c7caf2..35978b839 100644
--- a/src/dynamic_programming/knuth-optimization.md
+++ b/src/dynamic_programming/knuth-optimization.md
@@ -77,7 +77,7 @@ $$
 \sum\limits_{i=1}^N \sum\limits_{j=i}^{N-1} [opt(i+1,j+1)-opt(i,j)].
 $$
 
-As you see, most of the terms in this expression cancel each other out, except for positive terms with $j=N$ and negative terms with $i=1$. Thus, the whole sum can be estimated as
+As you see, most of the terms in this expression cancel each other out, except for positive terms with $j=N-1$ and negative terms with $i=1$. Thus, the whole sum can be estimated as
 
 $$
 \sum\limits_{k=1}^N[opt(k,N)-opt(1,k)] = O(n^2),